text: For laminar or turbulent jets and for laminar plumes, the volumetric entertainment rate per unit axial length is constant as can be seen from the solution of Schlichting jet and Yih plume. Thus, the jet or plume can be considered as a line sink that drives the motion in the outer region, as was first done by G. I. Taylor. Prior to Schneider, it was assumed that this outer fluid motion is also a large Reynolds number flow, hence the outer fluid motion is assumed to be a potential flow solution, which was solved by G. I. Taylor in 1958. For turbulent plume, the entrainment is not constant, nevertheless, the outer fluid is still governed by Taylors solution.
  Though Taylors solution is still true for turbulent jet, for laminar jet or laminar plume, the effective Reynolds number for outer fluid is found to be of order unity since the entertainment by the sink in these cases is such that the flow is not inviscid. In this case, full Navier-Stokes equations has to be solved for the outer fluid motion and at the same time, since the fluid is bounded from the bottom by a solid wall, the solution has to satisfy the non-slip condition. Schneider obtained a self-similar solution for this outer fluid motion, which naturally reduced to Taylors potential flow solution as the entrainment rate by the line sink is increased.
  Suppose a conical wall of semi-angle  with polar axis along the cone-axis and assume the vertex of the solid cone sits at the origin of the spherical coordinates  extending along the negative axis. Now, put the line sink along the positive side of the polar axis. Set this way,  represents the common case of flat wall with jet or plume emerging from the origin. The case  corresponds to jet/plume issuing from a thin injector. The flow is axisymmetric with zero azimuthal motion, i.e., the velocity components are . The usual technique to study the flow is to introduce the Stokes stream function  such that
  Introducing  as the replacement for  and introducing the self-similar form  into the axisymmetric Navier-Stokes equations, we obtain
  where the constant  is such that the volumetric entrainment rate per unit axial length is equal to . For laminar jet,  and for laminar plume, it depends on the Prandtl number , for example with , we have  and with , we have . For turbulent jet, this constant is the order of the jet Reynolds number, which is a large number.
  The above equation can easily be reduced to a Riccati equation by integrating thrice, a procedure that is same as in the Landau–Squire jet (main difference between Landau-Squire jet and the current problem are the boundary conditions). The boundary conditions on the conical wall  become
  and along the line sink , we have
  The problem has been solved numerically from here.